Problem: A group of adults and kids went to see a movie. Tickets cost $$8.50$ each for adults and $$4.50$ each for kids, and the group paid $$79.00$ in total. There were $6$ fewer adults than kids in the group. Find the number of adults and kids in the group.
Let $x$ equal the number of adults and $y$ equal the number of kids. The system of equations is then: ${8.5x+4.5y = 79}$ ${x = y-6}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${y-6}$ for $x$ in the first equation. ${8.5}{(y-6)}{+ 4.5y = 79}$ Simplify and solve for $y$ $ 8.5y-51 + 4.5y = 79 $ $ 13y-51 = 79 $ $ 13y = 130 $ $ y = \dfrac{130}{13} $ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into ${x = y-6}$ to find $x$ ${x = }{(10)}{ - 6}$ ${x = 4}$ You can also plug ${y = 10}$ into ${8.5x+4.5y = 79}$ and get the same answer for $x$ ${8.5x + 4.5}{(10)}{= 79}$ ${x = 4}$ There were $4$ adults and $10$ kids.